find the surface area of a regular polygon inscribed in a sphere in typescript

To find the surface area of a regular polygon inscribed in a sphere, we need to use some basic geometric formulas. Here's the code in TypeScript:

index.ts
function surfaceAreaOfPolygonInSphere(numSides: number, radius: number): number {
  const interiorAngle = (numSides - 2) * (180 / numSides);
  const apothem = radius * Math.cos(Math.PI / numSides);
  const sideLength = 2 * radius * Math.sin(Math.PI / numSides);
  const perimeter = numSides * sideLength;
  const surfaceArea = perimeter * apothem / 2;
  return surfaceArea;
}
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Here's how the code works:

1. The function takes two arguments: numSides is the number of sides of the regular polygon, and radius is the radius of the sphere.
2. We calculate the interior angle of the polygon using the formula (numSides - 2) * (180 / numSides).
3. Next, we calculate the apothem of the polygon using the formula radius * Math.cos(Math.PI / numSides). The apothem is the distance from the center of the polygon to the midpoint of a side.
4. We calculate the side length of the polygon using the formula 2 * radius * Math.sin(Math.PI / numSides).
5. We calculate the perimeter of the polygon by multiplying the number of sides by the side length: numSides * sideLength.
6. Finally, we calculate the surface area by multiplying the perimeter by the apothem and dividing by 2: perimeter * apothem / 2.

And that's it! Just call the function with the appropriate arguments to get the surface area of a regular polygon inscribed in a sphere.